A critical re-evaluation of the slope factor of the operational model of agonism: When to exponentiate operational efficacy

Agonist efficacy denoting the “strength” of agonist action is a cornerstone in the proper assessment of agonist selectivity and signalling bias. The simulation models are very accurate but complex and hard to fit experimental data. The parsimonious operational model of agonism (OMA) has become successful in the determination of agonist efficacies and ranking them. In 1983, Black and Leff introduced the slope factor to the OMA to make it more flexible and allow for fitting steep as well as flat concentration–response curves. First, we performed a functional analysis to indicate the potential pitfalls of the OMA. Namely, exponentiation of operational efficacy may break relationships among the OMA parameters. The fitting of the Black & Leff equation to the theoretical curves of several models of functional responses and the experimental data confirmed the fickleness of the exponentiation of operational efficacy affecting estimates of operational efficacy as well as other OMA parameters. In contrast, fitting The OMA based on the Hill equation to the same data led to better estimates of model parameters. In conclusion, Hill equation-based OMA should be preferred over the Black & Leff equation when functional-response curves differ in the slope factor. Otherwise, the Black & Leff equation should be used with extreme caution acknowledging potential pitfalls.


Power function
Exponentiation is a mathematical operation, written as b n , involving two numbers, the base b and the exponent n. b ≥ 0 can be risen to any value of n. b < 0 can be risen only to integers or fractions with odd denominators.Table S1 summarizes relationships between b n and b values ≥ 0. Reciprocal relationships apply for b values < 0 as the power function is symmetric at the origin of axes.Importantly relationship between values of b n and n are opposite in the range 0 < b < 1 to the range b > 1, resulting in the S-shape of the function (Figure S2).

Figure S2 Examples of power functions
Curves of y = x n for various values of exponent n.Each curve passes through the point (0, 0) because 0 raised to any power is 0 and through the point (1, 1) because the number 1 raised to any power is 1.For n = 1, y = x because any number raised to the power of 1 is the number itself.

Hyperbola and power function
Exponentiation of x in the hyperbola (Eq.A1) results in a non-hyperbolic function (Eq.A4, Figure S3).

Non-competitive auto-inhibition
In the case of non-competitive auto-inhibition, the functional response is given by Eq.S16.
Eq. S20 The apparent value of operational efficacy τ' is given by equation Eq.S21.

𝜏′ = 𝜏𝜎 𝜏+𝜎+1
For half-efficient concentration EC50: Eq. S23 Except for τ = 1, parameter estimates are correct, and associated with the low level of uncertainty.However, it is the result of the initial estimate of τ.For τ = 1 = σ, estimates are also correct but associated with a large uncertainty level as so far τ = σ curves given by Eq.S19 do not change.Parameter estimates are incorrect and for low values of τ and σ they are associated with high-level uncertainty.However, calculated τ and σ values give correct apparent efficacy according to Eq. S21.

Signalling feedback
In signalling feedback, an increase in output signal proportionally either decreases (negative feedback) or increases (positive feedback) input.Activation constant KE can be expressed as the difference between [RAG] formation and decay Eq. S 25.
Eq. S 25 Where [GT] is the total concentration of the effector.Eq. S 25 can be simplified to Eq. S 26.
Eq. S 26 After rearrangement Eq. S 27 The feedback factor δ modifies input [RA] Eq. S 28

Respectively
Eq. S 29 For δ > 1, the right member of Eq. S 28 is greater than the right member of Eq. S 26 and thus denotes attenuation of the signal, negative feedback.Conversely, δ < 1 denotes positive feedback.The proportion of signal output, [RAG], working as feedback is constant and is given by the division of Eq. S 29 by Eq. S 27.
Eq. S 30 Under the feedback, the response is given by: Eq. S 31 After simplification Eq. S 32 []+

Figure S6 Fitting Eq. S 37 to the signalling-feedback model with varying feedback
Dots, functional-response data modelled according to Eq. S 33.EMAX = 1, KA = 10 -6 M, KE = 0.333, RT = 1.Feedback factor δ varied from 0.2 to 5 and is indicated in the legend.Lines, fits of Eq. S 37 to the model data.EMAX was fixed to 1, the initial estimate of τ was set to 3 and the initial estimate of pKA was set to 6. Parameter estimates of the fits are indicated in the legend.

Figure S7 Fitting Eq. S 37 to the signalling-feedback model with constant feedback and varying operational efficacy
Dots, functional-response data modelled according to Eq. S 33.EMAX = 1, KA = 10 -6 M, RT = 1, δ=5.Operational efficacy varied from 0.2 to 5 and is indicated in the legend.Lines, fits of Eq. S 37 to the model data.EMAX was fixed to 1, the initial estimate of δ was set to 5 and the initial estimate of pKA was set to 6. Parameter estimates of the fits are indicated in the legend.

The system with a similar expression of receptor and effector ([RT]≈[GT])
Equation Eq.S6 is valid only when KE is given by: Eq.S43

Figure S 8 Modelling the system with a similar expression of receptor and effector ([RT]≈[GT])
Dots, functional-response data modelled in two steps.First, binding was calculated according to Eq. S5.Then resulting [RA] was used in Eq.S50.GT = 10 -6 M, RT = 10 -5 M, KA = 10 -6 M. Values of KE are indicated in the legend.As Eq.S52 gives only approximate solutions, data were not refitted.

The case study
Stimulation of [ 35 S]GTPγS binding to subtypes of inhibitory G-proteins upon activation of the M2 muscarinic receptor by agonists.

Python scripts
Scripts to model functional-response data and fit equations of Black & Leff, Hill and explicit models to them are provided in python.zip.

Figure
Figure S3 Examples of functions according to Eq. A4Curves according to Eq. A4 for various values of exponent n.Left, both branches are shown.Right, detail of right branches.Black, for the n=1, the curve is a rectangular hyperbola.Shades of blue, n < 1, and shades of yellow, n > 1, curves are S-shaped.

Figure S4 Fitting
Figure S4 Fitting Eq.S19 to the model of non-competitive auto-inhibitionDots, functional-response data modelled according to Eq. S17.EMAX = 1, KA = 10 -6 M, σ = 1, RT =1.Values of operational efficacy τ are indicated in the legend.Lines, fits of Eq.S19 to the model data.EMAX was fixed to 1, and the initial estimate of τ was set to 0.3.Parameter estimates of the fits are indicated in the legend.

Figure S5 Fitting
Figure S5 Fitting Eq.S19 to the model of non-competitive auto-inhibition with varying receptor concentrationDots, functional-response data modelled according to Eq. S17.EMAX = 1, KA = 10 -6 M, KE = 3.333, KI = 5.RT varied from 0.5 to 8. The resulting operational efficacies τ and inhibition factors σ are indicated in the legend.Lines, fits of Eq.S19 to the model data.EMAX was fixed to 1, and the initial estimate of τ was set to apparent operational efficacy τ' calculated as E'MAX/(EMAX-E'MAX).Parameter estimates of the fits are indicated in the legend.

Table S3 Results of fitting Black & Leff equations to the experimental data
Experimental data were normalized to maximal system response (Figure7).Black & Leff (Eq.7) and Hill(Eq.11)equations were fitted to the experimental data with EMAX fixed to 1. *, different from Black & Leff (p < 0.05) according to ANOVA and Tukey HSD post-test.